

Algorithm: Brute force conditional sampling and pricing

Fix the \(M\)  collocation points \(\xi \) .

Given the parameters \(\mathbf {p}\) , for \(j=1,\dots ,N_{\texttt {paths}}\) , repeat:

generate the sample \(\hat{S}_j\)  from \(S(T)\)  (e.g. with COS-SC method [1]} and [2]});

given \(\mathbf {p}_{\hat{S}_j}\) , approximate the \(M\)  conditional CVs, i.e. \(\mathbf {a}_{\hat{S}_j}\approx \tilde{H}(\mathbf {p}_{\hat{S}_j})\) ;

given \(\mathbf {a}_{\hat{S}_j}\) , use SC to generate the conditional sample \(\hat{A}_j\) .


Given the pairs \((\hat{S}_j,\hat{A}_j)\) , for \(j=1,\dots ,N_{\texttt {paths}}\) , and any desired \((K_1,K_2)\) , evaluate:
\(V_{\omega }(t_0)\approx \frac{1}{N_{\texttt {paths}}}\frac{M(t_0)}{M(T)}\sum _{j=1}^{N_{\texttt {paths}}}\max \big (\omega \big ({\hat{A}_j}-K_1\hat{S}_j-{K}_2\big ), 0\big ).\) 




